MSDS6373-Time Series-Air Quality

1 Abstract

Air pollution is one of the major concern in today’s world. It can have very serious cost, penalites and consequences for the health of human beings and also ruthlessly distresses natural bio-network and ecosystems.Out of five major air pollutants (round-level ozone, particle pollution also known as particulate matter, carbon monoxide, sulfur dioxide, and nitrogen dioxide) WHO has identified SPM (Suspended Particulated Matter) as most sinister in terms of its effects on health. You may experience health issues such as Irregular heartbeat,Respiratory symptoms like coughing, wheezing or difficulty breathing etc. Therefore measuring these pollutants on daily basis and forecast is vital for any government. Daily measuring and forecasts can help Governments in designing policies to control air pollution and issue a guidance to citizens of possible health impacts and precautions to be taken if air quality becomes bad or predicted to be bad on next day.

2 Intro to Dataset

With this little background, for academic purposes we have decided to use Air Quality dataset from UCI machine learning respository for time series term project. This data set includes hourly air pollutants data from 12 nationally-controlled air-quality monitoring sites. The air-quality data are from the Beijing Municipal Environmental Monitoring Center. The meteorological data in each air-quality site are matched with the nearest weather station from the China Meteorological Administration. The time period is from March 1st, 2013 to February 28th, 2017.

This hourly data set considers 6 main air pollutants and 6 relevant meteorological variables at multiple sites in Beijing.

SrNo	Air Pollutants	Description
1	PM2.5	PM2.5 concentration (ug/\(m^{3}\))
2	PM10	PM10 concentration (ug/\(m^{3}\))
3	SO2	Sulphur Dioxide concentration (ug/\(m^{3}\))
4	NO2	Nitrogen Dioxide concentration (ug/\(m^{3}\))
5	CO	Carbon Monoxide concentration (ug/\(m^{3}\))
6	O3	Ozone concentration (ug/\(m^{3}\))
7	TEMP	temperature (degree Celsius)
8	PRES	pressure (hPa)
9	DEWP	dew point temperature (degree Celsius)
10	RAIN	precipitation (mm)
11	wd	wind direction
12	WSPM	wind speed (m/s)

Goal
Goal of this project is to develop a model to forecast on all major air pollutants PM2.5,PM10,SO2,NO2,CO and O3 at least for next 10 days and effect of these pollutants on temperature and rain (Multivariate Analysis)

This dataset contains observations from 12 Chinese cities. For initial analysis (EDA) we are using observations from the city called Dongsi and it will be extended to two more cities for the purpose of the project and will try fit best possible model based on analysis.

Following are city names :

SrNo	City
1	Aoizhongxin
2	Changping
3	Dingling
4	Dongsi
5	Guanyuan
6	Gucheng
7	Huairou
8	Nongzhanguan
9	Shunyi
10	Titantan
11	Wanliu
12	Wanshouxigong

Let’s use data from the city called Dongsi.

c_city01_ds <- read.csv('data/PRSA_Data_Dongsi_20130301-20170228.csv')

3 Analysis of Dataset

3.1 Structure

str(c_city01_ds)

## 'data.frame':    35064 obs. of  18 variables:
##  $ No     : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ year   : int  2013 2013 2013 2013 2013 2013 2013 2013 2013 2013 ...
##  $ month  : int  3 3 3 3 3 3 3 3 3 3 ...
##  $ day    : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ hour   : int  0 1 2 3 4 5 6 7 8 9 ...
##  $ PM2.5  : num  9 4 7 3 3 4 5 3 3 3 ...
##  $ PM10   : num  9 4 7 3 3 4 5 6 6 6 ...
##  $ SO2    : num  3 3 NA 5 7 9 10 12 12 9 ...
##  $ NO2    : num  17 16 17 18 NA 25 29 40 41 31 ...
##  $ CO     : int  300 300 300 NA 200 300 400 400 500 400 ...
##  $ O3     : num  89 88 60 NA 84 78 67 52 54 69 ...
##  $ TEMP   : num  -0.5 -0.7 -1.2 -1.4 -1.9 -2.4 -2.5 -1.4 -0.3 0.4 ...
##  $ PRES   : num  1024 1025 1025 1026 1027 ...
##  $ DEWP   : num  -21.4 -22.1 -24.6 -25.5 -24.5 -21.3 -20.4 -20.4 -21.2 -23.3 ...
##  $ RAIN   : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ wd     : Factor w/ 16 levels "E","ENE","ESE",..: 7 8 7 4 7 8 8 7 8 4 ...
##  $ WSPM   : num  5.7 3.9 5.3 4.9 3.2 2.4 2.2 3 4.6 5.5 ...
##  $ station: Factor w/ 1 level "Dongsi": 1 1 1 1 1 1 1 1 1 1 ...

Combine Day Month and Year fields into single Date field.

c_city01_ds$Date <- as.Date(paste0(c_city01_ds[,2],
                                  Format(c_city01_ds[,3], digits=0, leading="00"),
                                  Format(c_city01_ds[,4], digits=0, leading="00")),'%Y%m%d')

c_city01_ds <- c_city01_ds[c(1,19,5:18)] %>% arrange(Date)
c_city01_ds

3.2 Missing Values

visdat::vis_miss(c_city01_ds)

3.3 Imputations

For imputation we will be using R package imputeTS . It has different methods depending on seasonality and trends to impute values for time series.

So let’s first plot the each series by excluding rows with NA values to check if there is trend or seasonality in the data and apply these methods [1] accordingly.

3.3.1 PM2.5 Series

Next Observartion Carried Backward Method

x <- seq(1, length(c_city01_ds$PM2.5), by = 1)
c_city01_ds$impute_pm25 <- na.locf(c_city01_ds$PM2.5, option = "nocb") 
c_city01_ds$indpm25     <- ifelse(is.na(c_city01_ds$PM2.5), c_city01_ds$impute_pm25, NA) 

fig <- plot_ly(x = ~x, y = ~c_city01_ds$PM2.5, name = 'PM2.5 Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indpm25, name = 'PM2.5 Imputed', type = 'scatter', mode = 'lines+markers')
fig <- fig %>% layout(title = "PM2.5 Series",
                      xaxis = list(title = "Time"),
                      yaxis = list (title = "Particulate Matter"))
fig

3.3.2 PM10 Series

Next Observartion Carried Backward Method

x <- seq(1, length(c_city01_ds$PM10), by = 1)
c_city01_ds$impute_pm10 <- na.locf(c_city01_ds$PM10, option = "nocb") 
c_city01_ds$indpm10     <- ifelse(is.na(c_city01_ds$PM10), c_city01_ds$impute_pm10, NA) 

fig <- plot_ly(x = ~x, y = ~c_city01_ds$PM10, name = 'PM10 Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indpm10, name = 'PM10 Imputed', type = 'scatter', mode = 'lines+markers')
fig <- fig %>% layout(title = "PM10 Series",
                      xaxis = list(title = "Time"),
                      yaxis = list (title = "Particulate Matter"))
fig

3.3.3 SO2 Series

Seasonal Adjustment then LOCF

x <- seq(1, length(c_city01_ds$SO2), by = 1)
c_city01_ds$impute_so2 <- na.seadec(c_city01_ds$SO2, algorithm = "locf")

## Warning: na.seadec will replaced by na_seadec.
##     Functionality stays the same.
##     The new function name better fits modern R code style guidelines.
##     Please adjust your code accordingly.

## Warning in na_seadec(x, algorithm, find_frequency, maxgap, ...): No seasonality information for dataset could be found, going on without decomposition.
##               Setting find_frequency=TRUE might be an option.

c_city01_ds$indso2     <- ifelse(is.na(c_city01_ds$SO2), c_city01_ds$impute_so2, NA) 

fig <- plot_ly(x = ~x, y = ~c_city01_ds$SO2, name = 'SO2 Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indso2, name = 'SO2 Imputed', type = 'scatter', mode = 'lines+markers')
fig <- fig %>% layout(title = "Sulphur Dioxide Series",
                      xaxis = list(title = "Time"),
                      yaxis = list (title = "Sulphur Dioxide"))
fig

3.3.4 NO2 Series

Next Observartion Carried Backward Method

x <- seq(1, length(c_city01_ds$NO2), by = 1)
c_city01_ds$impute_no2 <- na.locf(c_city01_ds$NO2, option = "nocb") 
c_city01_ds$indno2     <- ifelse(is.na(c_city01_ds$NO2), c_city01_ds$impute_no2, NA) 

fig <- plot_ly(x = ~x, y = ~c_city01_ds$NO2, name = 'NO2 Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indno2, name = 'NO2 Imputed', type = 'scatter', mode = 'lines+markers')
fig <- fig %>% layout(title = "Nitrogen Dioxide Series",
                      xaxis = list(title = "Time"),
                      yaxis = list (title = "Nitrogen Dioxide"))
fig

3.3.5 O3 Series

Seasonal Adjustment then LOCF

x <- seq(1, length(c_city01_ds$O3), by = 1)
c_city01_ds$impute_o3 <- na.seadec(c_city01_ds$O3, algorithm = "interpolation")

## Warning: na.seadec will replaced by na_seadec.
##     Functionality stays the same.
##     The new function name better fits modern R code style guidelines.
##     Please adjust your code accordingly.

## Warning in na_seadec(x, algorithm, find_frequency, maxgap, ...): No seasonality information for dataset could be found, going on without decomposition.
##               Setting find_frequency=TRUE might be an option.

c_city01_ds$indo3     <- ifelse(is.na(c_city01_ds$O3), c_city01_ds$impute_o3, NA) 

fig <- plot_ly(x = ~x, y = ~c_city01_ds$O3, name = 'O3 Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indo3, name = 'O3 Imputed', type = 'scatter', mode = 'lines+markers')
fig <- fig %>% layout(title = "Ozone Series",
                      xaxis = list(title = "Time"),
                      yaxis = list (title = "Ozone"))
fig

3.3.6 CO Series

Seasonal Adjustment then LOCF

x <- seq(1, length(c_city01_ds$CO), by = 1)
c_city01_ds$impute_co <- na.seadec(c_city01_ds$CO, algorithm = "interpolation")

## Warning: na.seadec will replaced by na_seadec.
##     Functionality stays the same.
##     The new function name better fits modern R code style guidelines.
##     Please adjust your code accordingly.

## Warning in na_seadec(x, algorithm, find_frequency, maxgap, ...): No seasonality information for dataset could be found, going on without decomposition.
##               Setting find_frequency=TRUE might be an option.

c_city01_ds$indco     <- ifelse(is.na(c_city01_ds$CO), c_city01_ds$impute_co, NA) 

fig <- plot_ly(x = ~x, y = ~c_city01_ds$CO, name = 'CO Original', type = 'scatter', mode = 'lines')
fig <- fig %>% add_trace(y = ~c_city01_ds$indco, name = 'CO Imputed', type = 'scatter', mode = 'lines+markers')
fig <- fig %>% layout(title = "Carbon Monoxide Series",
                      xaxis = list(title = "Time"),
                      yaxis = list (title = "Carbon Monoxide"))
fig

3.3.7 Imputed Dataset

ts_city01_ds <- c_city01_ds[c(1,2,3,17,19,21,23,25,27,10:16)]
visdat::vis_miss(ts_city01_ds)

3.3.8 Final Dataset

Final dataset for this time series analysis is obtained by converting hourly observations to averaged daily observations. This data will then be used to 10 day forecast.

ts_daily_df <- sqldf("select Date,avg(impute_pm25) PM25,avg(impute_pm10) PM10,
                      avg(impute_so2) SO2,avg(impute_no2) NO2,avg(impute_co) CO,
                      avg(impute_o3) O3 from ts_city01_ds group by Date order by Date")
ts_daily_df

4 PM2.5 Analysis

Following analysis is done by using tswge R package that we have used exstensively this course.

4.1 Stationarity

First plot the data.

plotts.sample.wge(ts_daily_df$PM25)

acf(ts_daily_df$PM25[1:730])
acf(ts_daily_df$PM25[731:1461])

4.2 Trend/Signal check

t <- seq(1,length(ts_daily_df$PM25),1)
fit_pm25 = lm(PM25~t, data = ts_daily_df)
summary(fit_pm25)

## 
## Call:
## lm(formula = PM25 ~ t, data = ts_daily_df)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -83.44 -52.83 -19.90  28.30 481.39 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 87.0021727  3.8605346  22.536   <2e-16 ***
## t           -0.0009465  0.0045744  -0.207    0.836    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 73.74 on 1459 degrees of freedom
## Multiple R-squared:  2.934e-05,  Adjusted R-squared:  -0.000656 
## F-statistic: 0.04281 on 1 and 1459 DF,  p-value: 0.8361

p-value > 0.05 (using OLS method) indicates b1 = 0 indicates no signal present in the data

4.3 Model ID

#aic5.wge(ts_daily_df$PM25,p=1:20,type='aic')

AR(1,1) turns out to be the best mode as per AIC function.

4.4 Estimate Parameters

4.4.1 Estimation

parms_pm25 <- est.arma.wge(ts_daily_df$PM25,p=5,q=1)

## 
## Coefficients of Original polynomial:  
## 1.5740 -0.7845 0.2155 -0.0550 0.0284 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-0.9633B              1.0381               0.9633       0.0000
## 1-0.8349B+0.2773B^2    1.5056+-1.1575i      0.5266       0.1043
## 1+0.2242B+0.1062B^2   -1.0558+-2.8813i      0.3259       0.3059
##   
## 

parms_pm25$phi

## [1]  1.57395167 -0.78450207  0.21553552 -0.05495932  0.02836324

parms_pm25$theta

## [1] 0.933258

mean(ts_daily_df$PM25)

## [1] 86.31029

pacf(parms_pm25$res)

4.4.2 Residual diagnositcs

ljung.wge(parms_pm25$res,p=5,q=1)

## Obs -0.0002931233 -0.001050439 -0.002165653 -0.0002722587 0.006506776 -0.01155279 -0.0435071 -0.005600042 0.01912308 0.01538431 0.01577499 -0.02114577 0.00706101 0.01408816 0.027602 0.004925593 -0.01359664 -0.01152449 -0.01264407 -0.02226326 0.02887934 0.02193741 -0.02797961 0.03644576

## $test
## [1] "Ljung-Box test"
## 
## $K
## [1] 24
## 
## $chi.square
## [1] 13.07265
## 
## $df
## [1] 18
## 
## $pval
## [1] 0.7872393

4.5 Final Model

Final PM2.5 Model:
(1 - 1.58B + 0.79\(B^2\) - 0.22\(B^3\) + 0.06\(B^4\) - 0.03\(B^5\)) (\(X_t\) - 85.96) = (1 - 0.93B) (\(a_t\))

4.6 PM2.5 Forecasting

fore_pm25 <- fore.arma.wge(ts_daily_df$PM25,phi=parms_pm25$phi,theta=parms_pm25$theta, n.ahead=20,lastn=TRUE)

x <- seq(1350, 1461, by = 1)
y <- ts_daily_df$PM25[1350:1461]
fx <- seq(1442, 1461, by = 1) 

fig <- plot_ly()
fig <- fig %>% add_lines(x = x, y = y,
                color = I("black"), name = "observed")
fig <- fig %>% add_lines(x = fx, y = fore_pm25$f, color = I("blue"), name = "prediction")
fig <- fig %>% add_ribbons(x = fx, ymin = fore_pm25$ll, ymax = fore_pm25$ul,
               color = I("red"), name = "95% confidence")
fig <- fig %>% layout(title = "PM2.5 Forecast ",
                      xaxis = list(title = "Time"),
                      yaxis = list (title = "PM2.5"))

fig

4.7 ASE

ASE_pm25 <- mean((fore_pm25$f-ts_daily_df$PM25[1442:1461])^2)
ASE_pm25

## [1] 3523.562

5 PM10 Analysis

5.1 Stationarity

First plot the data.

plotts.sample.wge(ts_daily_df$PM10)

acf(ts_daily_df$PM10[1:730])
acf(ts_daily_df$PM10[731:1461])

5.2 Trend/Signal check

t <- seq(1,length(ts_daily_df$PM10),1)
fit_pm10 = lm(PM10~t, data = ts_daily_df)
summary(fit_pm10)

## 
## Call:
## lm(formula = PM10 ~ t, data = ts_daily_df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -105.68  -58.99  -19.56   32.07  473.75 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.102e+02  4.208e+00   26.19   <2e-16 ***
## t           5.000e-04  4.986e-03    0.10     0.92    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 80.38 on 1459 degrees of freedom
## Multiple R-squared:  6.894e-06,  Adjusted R-squared:  -0.0006785 
## F-statistic: 0.01006 on 1 and 1459 DF,  p-value: 0.9201

p-value > 0.05 (using OLS method) indicates b1 = 0 indicates no signal present in the data

5.3 Model ID

#aic5.wge(ts_daily_df$PM10,p=1:20,q=1:4,type='aic')

AR(6,1) turns out to be the best mode as per AIC function.

5.4 Estimate Parameters

5.4.1 Estimation

parms_pm10 <- est.arma.wge(ts_daily_df$PM10,p=6,q=1)

## 
## Coefficients of Original polynomial:  
## 1.5341 -0.7206 0.1976 -0.0402 -0.0373 0.0481 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-0.9717B              1.0291               0.9717       0.0000
## 1-1.0950B+0.4195B^2    1.3052+-0.8248i      0.6477       0.0897
## 1+0.1225B+0.2875B^2   -0.2130+-1.8528i      0.5362       0.2682
## 1+0.4101B             -2.4383               0.4101       0.5000
##   
## 

parms_pm10$phi

## [1]  1.53414843 -0.72055103  0.19758558 -0.04024130 -0.03726040  0.04806275

parms_pm10$theta

## [1] 0.9325623

mean(ts_daily_df$PM10)

## [1] 110.5843

pacf(parms_pm10$res)

5.4.2 Residual diagnositcs

ljung.wge(parms_pm10$res,p=6,q=1)

## Obs -0.00104693 -0.002345418 0.0006287611 -0.001048871 0.005649315 0.004902281 -0.03849331 -0.002147024 0.005807087 0.04055073 -0.01048374 -0.01530926 0.00174568 0.01310702 0.01360484 0.002005844 -0.01765535 -0.046519 -0.01001625 -0.02519184 0.007243046 0.03297038 -0.01377204 0.04518259

## $test
## [1] "Ljung-Box test"
## 
## $K
## [1] 24
## 
## $chi.square
## [1] 15.5644
## 
## $df
## [1] 17
## 
## $pval
## [1] 0.5548968

5.5 Final Model

Final PM10 Model:

(1 - 1.53B + 0.72\(B^2\) - 0.18\(B^3\) + 0.04\(B^4\) + 0.03\(B^5\) - 0.05\(B^5\) ) (\(X_t\) - 110.111) = (1 - 0.93B) (\(a_t\))

5.6 PM10 Forecasting

fore_pm10 <- fore.arma.wge(ts_daily_df$PM10,phi=parms_pm10$phi,theta=parms_pm10$theta, n.ahead=20,lastn=TRUE)

x <- seq(1350, 1461, by = 1)
y <- ts_daily_df$PM10[1350:1461]
fx <- seq(1442, 1461, by = 1) 

fig <- plot_ly()
fig <- fig %>% add_lines(x = x, y = y,
                color = I("black"), name = "observed")
fig <- fig %>% add_lines(x = fx, y = fore_pm10$f, color = I("blue"), name = "prediction")
fig <- fig %>% add_ribbons(x = fx, ymin = fore_pm10$ll, ymax = fore_pm10$ul,
               color = I("red"), name = "95% confidence")
fig <- fig %>% layout(title = "PM10 Forecast ",
                      xaxis = list(title = "Time"),
                      yaxis = list (title = "PM10"))

fig

factor.wge(parms_pm10$phi)

## 
## Coefficients of Original polynomial:  
## 1.5341 -0.7206 0.1976 -0.0402 -0.0373 0.0481 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-0.9717B              1.0291               0.9717       0.0000
## 1-1.0950B+0.4195B^2    1.3052+-0.8248i      0.6477       0.0897
## 1+0.1225B+0.2875B^2   -0.2130+-1.8528i      0.5362       0.2682
## 1+0.4101B             -2.4383               0.4101       0.5000
##   
## 

factor.wge(parms_pm10$theta)

## 
## Coefficients of Original polynomial:  
## 0.9326 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-0.9326B              1.0723               0.9326       0.0000
##   
## 

5.7 ASE

ASE_pm10 <- mean((fore_pm10$f-ts_daily_df$PM10[1442:1461])^2)
ASE_pm10

## [1] 4542.258

6 SO2 Analysis

6.1 Stationarity

First plot the data.

plotts.sample.wge(ts_daily_df$SO2)

acf(ts_daily_df$PM10[1:730])
acf(ts_daily_df$PM10[731:1461])

Mean does not depend on time

Realization shows strong evidence of seasonality. ACFs are slowly damped and there are few peaks in spectral density plots. Peaks are seen at 0, 0.13(Weekly). Dips are at 0.08,0.18,0.38 and 0.45. Although except peak at 0 and 0.13 other are weak evidences of frequency in the data.

Variance does not depend on time

correlation between \(X_t_1}\) \(X_t_2}\) depends on \(t_2\)- \(t_1\) and not where they are in time

ACFs on two equal half portion of the data shows correlation depends on time and therefore it violates this condition.

Therefore this series is not astationary.

6.2 Trend/Signal check

t <- seq(1,length(ts_daily_df$SO2),1)
fit_so2 = lm(SO2~t, data = ts_daily_df)
summary(fit_so2)

## 
## Call:
## lm(formula = SO2 ~ t, data = ts_daily_df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -25.129 -12.442  -5.715   6.298 115.679 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 28.244311   0.994978   28.39   <2e-16 ***
## t           -0.013293   0.001179  -11.28   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 19.01 on 1459 degrees of freedom
## Multiple R-squared:  0.08015,    Adjusted R-squared:  0.07952 
## F-statistic: 127.1 on 1 and 1459 DF,  p-value: < 2.2e-16

cfit_so2 = cochrane.orcutt(fit_so2) 
summary(cfit_so2)

## Call:
## lm(formula = SO2 ~ t, data = ts_daily_df)
## 
##               Estimate Std. Error t value  Pr(>|t|)    
## (Intercept) 28.3959630  2.0767594  13.673 < 2.2e-16 ***
## t           -0.0134533  0.0024557  -5.478 5.052e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.8334 on 1458 degrees of freedom
## Multiple R-squared:  0.0202 ,  Adjusted R-squared:  0.0195
## F-statistic: 30 on 1 and 1458 DF,  p-value: < 5.052e-08
## 
## Durbin-Watson statistic 
## (original):    0.74940 , p-value: 5.398e-127
## (transformed): 2.01091 , p-value: 5.723e-01

p-value < 0.05 (using Cochrane-Orcutt Test) provides strong evidence that trend exist.

6.3 Model ID

#aic5.wge(ts_daily_df$SO2,p=0:30,q=0:6,type='bic')

AR(9,1) turns out to be the best mode as per AIC function.

6.4 Estimate Parameters

6.4.1 Estimation

parms_so2 <- est.arma.wge(ts_daily_df$SO2,p=9,q=1)

## 
## Coefficients of Original polynomial:  
## 1.4029 -0.5464 0.0795 -0.0164 0.1073 -0.0848 -0.0244 0.2167 -0.1419 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-0.9908B              1.0093               0.9908       0.0000
## 1-1.2552B+0.6852B^2    0.9160+-0.7877i      0.8277       0.1130
## 1-0.2310B+0.6437B^2    0.1794+-1.2335i      0.8023       0.2270
## 1+1.0585B+0.6216B^2   -0.8514+-0.9401i      0.7884       0.3671
## 1+0.7307B             -1.3685               0.7307       0.5000
## 1-0.7152B              1.3983               0.7152       0.0000
##   
## 

d1_daily_so2 <-artrans.wge(ts_daily_df$SO2,phi.tr = 1)

d1_est_so2 = est.arma.wge(d1_daily_so2,p=9,q=1)

## 
## Coefficients of Original polynomial:  
## 0.4225 -0.1330 -0.0516 -0.0683 0.0401 -0.0463 -0.0702 0.1453 0.0051 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-1.2593B+0.6901B^2    0.9123+-0.7853i      0.8307       0.1131
## 1-0.2364B+0.6462B^2    0.1829+-1.2304i      0.8039       0.2265
## 1+1.0516B+0.6179B^2   -0.8510+-0.9456i      0.7860       0.3666
## 1-0.7387B              1.3536               0.7387       0.0000
## 1+0.7255B             -1.3783               0.7255       0.5000
## 1+0.0348B             -28.7668               0.0348       0.5000
##   
## 

d1_est_so2$phi

## [1]  0.422529679 -0.133032504 -0.051579252 -0.068347519  0.040095263
## [6] -0.046341580 -0.070162449  0.145306647  0.005133966

d1_est_so2$theta

## [1] 0.907354

pacf(d1_est_so2$res)
ljung.wge(d1_est_so2$res,p=9,q=1)

## Obs -0.001752766 0.005348165 -0.00141413 0.0006021668 -0.003601681 0.00736844 -0.003218477 0.0006892955 0.01216199 -0.01767099 -0.01022156 -0.02613282 0.05670951 -0.02612948 0.05192423 -0.04315283 -0.04373766 0.01366933 -0.04351761 -0.004347992 -0.006651504 0.01523455 0.03650025 -0.002965029

## $test
## [1] "Ljung-Box test"
## 
## $K
## [1] 24
## 
## $chi.square
## [1] 22.83122
## 
## $df
## [1] 14
## 
## $pval
## [1] 0.06307211

6.4.2 Residual diagnositcs

ljung.wge(parms_so2$res,p=2,q=2)

## Obs -0.003239577 0.002701459 0.001243839 -0.005176032 -0.008703925 0.003541291 -0.004860781 -0.002254335 0.01461634 -0.02273508 -0.01016355 -0.02621313 0.05318427 -0.02956708 0.05061178 -0.04635682 -0.0420046 0.01542838 -0.04253254 -0.0039102 -0.005211346 0.0171068 0.03978618 -0.002457134

## $test
## [1] "Ljung-Box test"
## 
## $K
## [1] 24
## 
## $chi.square
## [1] 23.41837
## 
## $df
## [1] 20
## 
## $pval
## [1] 0.2687217

6.5 Final Model

Final SO2 Model:

(1 - 1.40B + 0.558\(B^2\) - 0.088\(B^3\) + 0.028\(B^4\) - 0.118\(B^5\) + 0.088\(B^6\) + 0.028\(B^7\) - 0.218\(B^8\) + 0.148\(B^2\))(\(X_t\) - 18.53) = (1 - 0.89B) (\(a_t\))

6.6 SO2 Forecasting

fore_so2 <- fore.arma.wge(ts_daily_df$SO2,phi=d1_est_so2$phi,theta=d1_est_so2$theta, n.ahead=20,lastn=TRUE)

x <- seq(1350, 1461, by = 1)
y <- ts_daily_df$SO2[1350:1461]
fx <- seq(1442, 1461, by = 1) 

fig <- plot_ly()
fig <- fig %>% add_lines(x = x, y = y,
                color = I("black"), name = "observed")
fig <- fig %>% add_lines(x = fx, y = fore_so2$f, color = I("blue"), name = "prediction")
fig <- fig %>% add_ribbons(x = fx, ymin = fore_so2$ll, ymax = fore_so2$ul,
               color = I("red"), name = "95% confidence")
fig <- fig %>% layout(title = "SO2 Forecast ",
                      xaxis = list(title = "Time"),
                      yaxis = list (title = "Sulphur Dioxide"))
fig

## ASE

ASE_so2 <- mean((fore_so2$f-ts_daily_df$SO2[1442:1461])^2)
ASE_so2

## [1] 178.5033

factor.wge(d1_est_so2$phi)

## 
## Coefficients of Original polynomial:  
## 0.4225 -0.1330 -0.0516 -0.0683 0.0401 -0.0463 -0.0702 0.1453 0.0051 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-1.2593B+0.6901B^2    0.9123+-0.7853i      0.8307       0.1131
## 1-0.2364B+0.6462B^2    0.1829+-1.2304i      0.8039       0.2265
## 1+1.0516B+0.6179B^2   -0.8510+-0.9456i      0.7860       0.3666
## 1-0.7387B              1.3536               0.7387       0.0000
## 1+0.7255B             -1.3783               0.7255       0.5000
## 1+0.0348B             -28.7668               0.0348       0.5000
##   
## 

factor.wge(d1_est_so2$theta)

## 
## Coefficients of Original polynomial:  
## 0.9074 
## 
## Factor                 Roots                Abs Recip    System Freq 
## 1-0.9074B              1.1021               0.9074       0.0000
##   
## 

7 Conclusion

• Forecast from ARMA model are not correct forecast
• AR(1) is the strongest behavior in the above forecast. Doesn’t really follow the trend and Seasonality.
• Multivariate analysis is required on this dataset
• PM2.5 and PM10 forecasts may be well explained if CO,SO2 and O3 are taken into account.

8 What Next ?

Final part of the project will include following

• Multivariate Analysis (include all pollutants)
• Effect of pollutants on Rain and Temperature
• Develop models with Neural Network.

9 References

• Applied Time Series with R, Dr.Wayne Woodword,Henry L. Gray,Alan C. Elliott
• An article on ENVIRONMENTAL EFFECTS OF AIR POLLUTION AND APPLICATION OF ENGINEERED METHODS TO COMBAT THE PROBLEM, Journal of Industrial Pollution Control.
• Middle Tennessee Air Quality Forecast , Nashville Health Department.
• An article on Major Air pollutants.
• An article on common Air Pollutants and its effects on human beings.
• World Air Quality Forecast
• Kaggle article on handling Time series missing values. [1]